RDCH 702: Lecture 3, Nuclear Reactions
• Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and
Radiochemistry, Chapter 4
• Notation
• Energetics of Nuclear Reactions
• Reaction Types and Mechanisms

Barriers

Scattering
• Nuclear Reaction Cross Sections
• Reaction Observables
• Scattering
• Direct Reactions
• Compound Nuclear Reactions
• Photonuclear Reactions
• Nucleosynthesis
3-1
Nuclear Reactions
•
•
Nucleus reactions with a range of particles

Nucleus, subatomic particle, or photon to produce other nuclei

Short time frame (picosecond)

Energetics involved in reaction
First nuclear reaction from Rutherford

What reaction was this?
Number of terms conserved during nuclear reactions

Basis for understanding and evaluating reactions
 Number of nucleons
* except in reactions involving creation or annihilation of antinucleons
 Charge
 Energy/Mass
 Momentum
 Angular momentum
 Parity
Q is the energy of the reaction
•
 positive Q corresponds to energy release
 negative Q to energy absorption
Q terms given per nucleus transformed
•
•
14
7
N  He O  H  Q
4
2
Shorthand:
17
8
14
1
1
N ( , p) O
17
3-2
Energetics
• Energetically many orders of magnitude greater
than chemical reactions
• 14N(,p)17 O; Q=-1.193 MeV
 Convert energy to per molar basis
1 eV = 1.60E-19 J
=-7.18E23 MeV/mole= -7.18E29 eV/mole
=-1.15E11 J/mole
• Reactions so large that mass change is observable
• Q value can be experimentally measured to provide
a route to determine particle mass of reactants
 Mass and energy balance
Know Q value, determine unknown mass
3-3
Energetics
• Reaction Q values
 Not necessarily equal to kinetic energy of bombarding particles
for the reaction to occur
 Need more energy than Q value for reaction to occur
* Reaction products will have kinetic energy that needs to
come from reaction
• Conservation of momentum
 Some particles’ kinetic energy must be retained by products as
kinetic energy
• Amount retained as kinetic energy of products
 Based on projectile mass
 Retained kinetic energy becomes smaller with increasing target
mass
APr ojectile
Equation for kinetic energy (T):
T
Q
• What does this mean about reaction
APr ojectile  AT arg et
 Heavier target or heavier projectile?
248Cm + 18O266Rf
18
248
T
Q  0.932Q
248  18
T
248Cm
Projectile
248  18
18O
Q  0.068Q
Projectile
3-4
Energetics: Reaction Barrier
•
•
Need to consider laboratory and center of mass
frame
Laboratory frame

conservation of momentum considers
angle of particles
mp
mx
2
Q  Tx (1 
)  Tp (1 
)
(m pTp mxTx ) cosq
mR
mR mR
•
•
•
•
Q value can be found if Tx and q are measured
and particles known

Tp from experiment
Center of mass

Total particle angular momentum is zero
v p mp
2
(m p  mT )vcm
vcm 
Tcm 
(m p  mT )
2
Kinetic energy carried by projectile (Tlab) is not
fully available for reaction

Tlab - Tcm = T0

T0 is energy to be dissipated in reaction
For reaction to occur Q + T0 must be achieved

Basis for threshold reaction

Q + T0 > 0
Q  Tx  Tp  TR
Tcm  Tlab (
mp
m p  mT
)
3-5
•
Reaction Barrier
Threshold energy (minimum energy for reaction)
Q  Tlab  TCM  0; Tcm  Tlab (
Tlab  Tlab (
Tlab (1  (
Tlab 
m p  mT
mp
m p  mT
m p  mT
)
Solve of laboratory T
)  Q
))  Q
Q
Q
Q


mp
m p  mT
mp
mT
(1  (
)) (
(
))
m p  mT
m p  mT
m p  mT
m p  mT
T  Q
•
mp
mp
APr ojectile  AT arg et
AT arg et
A for mass
MeV
Fraction of bombarding particle’s kinetic energy retained as kinetic energy of
products becomes smaller with increasing mass of target

Heavier target or heavier projectile?
248Cm + 18O266Rf

3-6
Reaction Barrier: Threshold Energy
•
Consider the 14N(,p)17O reaction
A
 AT arg et

Find threshold energy
T  Q Pr ojectile
MeV
 Q from mass excess
AT arg et
* Q=2.425 + 2.863 – 7.289 – (-0.809) = -1.19 MeV
T  ( )1.19
•
•
•
•
4  14
MeV  1.53MeV
14
Reaction barrier also induced by Coulomb interaction

Need to have enough energy to react and overcome Coulomb barrier
 From charge repulse as particle approach each other
* R is radius
Z1 Z 2 e 2
Vc 
R  ro A1 / 3
* ro =1.1 to 1.6 fm
R1  R2
Equation can vary due to ro
Z1 Z 2
Vc can be above threshold energy
Vc  0.96 1 / 3
MeV
1/ 3
2*7
Vc  0.96 1 / 3
MeV  3.36 MeV
1/ 3
4  14
A1  A2
Center of mass, need to bring to laboratory frame

Consider kinetic energy carried by projectile

3.36x ((14+4)/14) = 4.32 MeV alpha needed for reaction
3-7
Equations for production reactions:
Cross Sections
• Probability of a nuclear process is generally
expressed in terms of a cross section 
 dimensions of an area
• Originates from probability for reaction between
nucleus and impinging particle is proportional to the
cross-sectional target area presented by the nucleus
 Doesn’t hold for charged particles that have to
overcome Coulomb barriers or for slow neutrons
• Total cross section for collision with fast particle is
never greater than twice the geometrical crosssectional area of the nucleus
 cross section  is close to 1 barn for this case
• 10-24 cm2=1 barn
3-8
Cross sections
• Accelerator: beam of particles striking a thin target with
minimum beam attenuation
Ri  Inx i
• When a sample is embedded in a uniform flux of particles
incident on it from all direction, such as in a nuclear reactor,
the cross section is defined:
Ri  N i

Ri= # of processes of type under consideration
occurring in the target per unit time
 I= # of incident particles per unit time
 n= # of nuclei/cm3
 x=target thickness (cm)
 =flux of particles/cm2/sec
 N=number of nuclei contained in sample
3-9
Production of radionuclides
N 0
=cross section
N1 
(1  e l1t )
l1
=neutron flux
l1t
N
l

A

N

(
1

e
)
t=time of irradiation
1 1
1
0
(1-e-(lt))
* maximum level (saturation factor)
• Activity of radioactive product at end bombardment
is divided by saturation factor, formation rate is
obtained
 R=A/(1-e-(lt))



half life
%
1
50
2
75
3
87.5
4
93.75
5
96.875
3-10
Nuclei production: Short irradiation
compared to half-life
• Find amount of 59Fe (t1/2=44.5 d, l = 1.803E-7 s-1) from
irradiation of 1 g of Fe in a neutron flux of 1E13 n/cm2/s for
1 hour
 58Fe(n,g)59Fe: 58Fe+ n g + 59Fe 1.3E-24 cm2
 No= 1g/55.845 g/mol *6.02E23 atom/mol*0.00282
 No=3.04E19 atom
• R= 1E13 n/cm2/s *1.3E-24 cm2 * 3.04E21 atom
• R=3.952E8 atoms/sec
Ri  N i
• 1.423E12 atoms 59Fe in 1 hour
3.04E19(1.3E - 24)(1E13)
N1 
(1  e 1.803E-7*3600 )
1.803E- 7
N 0
3.952E8
N1 
(1  e l1t )
N1 
(1  9.994E  1)
l1
1.803E- 7
N1  2.192E15(6.489E  4)  1.422E12 atom s
3-11
Nuclei production: Long irradiation
compared to half-life
• Find amount of 56Mn (t1/2=2.578 hr, l = 7.469E-5 s-1) from
irradiation of 1 g of Mn in a neutron flux of 1E13 n/cm2/s for
1 hour
 55Mn(n,g)56Mn: 55Mn+ n g + 56Mn 13.3E-24 cm2
 No= 1g/54.93804 g/mol *6.02E23 atom/mol
 No=1.096E22 atom
• R= 1E13 n/cm2/s *13.3E-24 cm2 * 1.096E22 atom Ri  N i
• R=1.457E12 atoms/sec
• 5.247E15 atoms 56Mn in 1 hour (does not account for decay)
N 0
l1t
1.096e22(
13.3E- 24)(1E13)
 7.469E -5*3600
N

(
1

e
)
N1 
(1  e
) 1
l1
7.469E- 5
1.458E12
N1 
(1  7.642E  1)
7.469E- 5
N1  1.952E16(2.358E  1)  4.603E15 atom s
3-12
Formation rate from activity
• R=A/(1-e-(lt))
• 4.603E15 atoms 56Mn (t1/2=2.578 hr, l = 7.469E5 s-1) from 1 hour irradiation
• A=lN= 4.603E15* 7.469E-5 =3.436E11 Bq
• R=A/(1-e-(lt))
• R= 3.436E11/(1-exp(- 7.469E-5 *3600))
• R=1.457E12 atom/sec
3-13
Cross Section Values and Limits
• Reaction cross section of R2 is approximated at high energies
 Wave nature of incident particle causes upper limit of reaction
cross section to include de Broglie wavelength
So cross section can be larger than area due to incoming
particle wavelength
Expressed as an increase in R, quantum in nature
 r   ( R  )
2
• Collision between neutron and target nucleus characterized by
distance of closest approach
 B is impact parameter
3-14
Cross sections
• Angular momentum of system is normal to the relative
momentum p
b
L  pb 
 l

b  l
• b any value between 0 and R
l  b  (l  1)
• l =0,1,2,…b angular momentum
2
 r   ( R  )
 lħ
• Sum all l from 0 to lmax
• Cross section based on summation of l cross sections
• For this reason nuclear reaction cross sections can be
several orders of magnitude larger than the nuclear
geometrical cross section
 Manifest by slow-neutron reactions
3-15
Cross section
l is partial cross section
of given angular
momentum l
 l   [(l 1)  l ]   (2l 1)
2
•
2
2
2
Quantum-mechanical treatment Tl is the
transmission coefficient for reaction of a
neutron with angular momentum l

Represents fraction of incident
particles with angular momentum l
that penetrate within range of
nuclear forces
 Provides summing term to
increase cross section
 Reason why cross section can be
larger than physical size of
nucleus
 r  

2
 (2l  1T
l 0
l
•
General trends for neutron and
charged particles

Charged particle cross section
minimal at low energy

Neutron capture cross section
maximum at low energy
3-16
•
•
Measuring Cross Section: Excitation Functions
Variation of reaction cross section with incident energy
Shape can be determined by exposing several target foils in same beam with energydegrading
 Simultaneous measurement of multiple particle energies
• Provide information about probabilities for emission of various kinds and combination of
particles in nuclear reactions
 formation of given product implies what particles were ejected from target nuclide
• Range of cross sections can be evaluated
 Detection limit of product can influence cross section limit measurement
3-17
Barriers for Charged Particles
• Coulomb repulsion between charged
bombarding particles and nucleus
 Repulsion increases with
decreasing distance of separation
until charged particle comes
within range of nuclear forces
 Probability of tunneling through
barrier drops rapidly as energy
of particle decreases
 Coulomb barriers affect charged
particles both entering and
leaving the nucleus
Charged particles emitted
from nuclei experience
Coulomb repulsion during
emission
greater than 1 MeV
seen with position emission
• Related to change in cross section
with energy for charged particle
reactions
 Maximum cross section
dependent upon energy
3-18
RDCH 702: Lecture 3, Nuclear Reactions
• Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and
Radiochemistry, Chapter 4
• Notation
• Energetics of Nuclear Reactions
• Reaction Types and Mechanisms

Barriers

Scattering
• Nuclear Reaction Cross Sections
• Reaction Observables
• Scattering
• Direct Reactions
• Compound Nuclear Reactions
• Photonuclear Reactions
• Nucleosynthesis
3-19
Reactions: Elastic Scattering
• Elastic scattering
 kinetic energy conserved
 Particles do not change
• Simplest consequence of a nuclear collision
 Not a “reaction”
no exchange of nucleons or creation of particles
• Particles do not change their identity during the process and
the sum of their kinetic energies remains constant
• Elastic scattering will also have a contribution from nuclear
forces
3-20
•
Low-Energy Reactions with Light
Slow-Neutron Reactions Projectiles
 Purest example of
compound-nucleus
behavior
1/v law governs most
neutron cross sections
in region of thermal
energies
 neutrons available only
from nuclear reactions
Range of energies can
be obtained
• Reaction Cross Sections
 Coulomb barrier prevents
study of nuclear reactions
with charged particles
below 1 MeV
resonances no longer
observable
with increasing energy,
increasing variety of
reactions possible
3-21
Low-Energy Reactions
• Deuteron Reactions
 Prevalence of one nucleon stripping
large size and loose binding of deuteron
Only proton and neutron in deuteron nucleus
* Proton charge carries both nucleons
 Neutron comes within range of nuclear forces while proton is still
outside most of Coulomb barrier
Inherent in large neutron-proton distance in deuteron
weakly bound deuteron can be broken up
* proton outside barrier
• Competition among Reactions
 depends on relative probabilities for emission of various particles
from compound nucleus
determined by number of factors
* energy available
* Coulomb barrier
* density of final states in product nucleus
3-22
High Energy Reactions
•
•
Spallation Products
 products in immediate neighborhood of target element found in
highest yields
 within 10 to 20 mass numbers
 yields tend to form in two regions
  stability for medium-weight products
 neutron-deficient side of stability with increasing Z of products
 Used to produce beam of neutrons at spallation neutron source
 Heavy Z will produce 20-30 neutrons
 Basis of Spallation neutron source
(http://neutrons.ornl.gov/facilities/SNS/)
High-Energy Fission
 single broad peak in mass-yield curve instead of double hump seen
in thermal-neutron fission
 many neutron-deficient nuclides
 especially among heavy products
 originate from processes involving higher deposition energies
 lower kinetic energies
 do not appear to have partners of comparable mass
 arise from spallation-like or fragmentation reactions
3-23
High-Energy Reactions
•
•
Mass-Yield Curves
 at low energies, compound-nucleus picture dominates
 as energy increases importance of direct reactions and preequilibrium (pre-compound nucleus)
emission increase
 above 100 MeV, nuclear reactions proceed nearly completely by direct interactions
 products down to mass number 150 are spallation products
 those between mass numbers 60 and 140 are fission products
Cascade-Evaporation Model

Above 100 MeV reactions

energy of the incident proton larger than interaction energy between the nucleons in the nucleus

Wavelength less than average distance between nucleons
 proton will collide with one nucleon at a time within the nucleus
* high-energy proton makes only a few collisions in nucleus
* Produces nucleons with high energy
3-24
Heavy-Ion Reactions
• Range of heavy ion reactions
 elastic and inelastic scattering
 compound-nucleus formation,
 direct interactions
 deeply inelastic reaction
• Reactions influence by parameter
 impact parameter of collision
 kinetic energy of projectile
1/ 3
1/ 3
R  ro A1  A2
 masses of target
 projectile nuclei
• Elastic and Inelastic Scattering, Coulomb Excitation
 elastic-scattering measurements used to obtain
information on interaction radii
 R=R1+R2 between mass numbers A1 and A2
(

3-25
Heavy Ion Reactions
• Inelastic scattering
 scattering in which some of projectile’s kinetic energy
transformed into excitation of target nucleus
greatest importance at large impact parameters
 heavy ions valuable
can excite high-spin states in target nuclei because of
large angular momenta
• Can experience Coulomb excitation
 high charges
 below Coulomb barrier heights and excite nuclei by purely
electromagnetic interactions
• Transfer Reactions
 stripping and pickup reactions prevalent with heavy ions
take place at impact parameters just below those at
which interactions are purely Coulombic
 angular distributions show oscillatory, diffraction-like
pattern when transfer reaction to single, well-defined state
observed
3-26
Heavy Ion Reactions: Deep Inelastic Reactions
• Relatively large amounts of nuclear matter
transferred between target and projectile
 Show strongly forward-peaked angular
distributions
 “Grazing contact mechanism”
• Products with masses in vicinity of projectile mass
appear at angles other than classical grazing angle
 Relatively small kinetic energies
• Total kinetic energies of products strongly correlated
with amount of mass transfer
 Increasing mass difference of product and
projectile lowers kinetic energy
• Product will dissociate into two fragments
 Appreciable fraction of incident kinetic energy
dissipated and goes into internal excitation
3-27
Compound-Nucleus Reactions
• Compound-nucleus formation can
only take place over a restricted range
of small impact parameters
 can define critical angular
momentum above which
complete fusion cannot occur
 cf/R decreases with increasing
bombarding energy
• Neutron deficient heavy ions produce
compound nuclei on neutron-deficient
side of  stability belt
• Heavy ion of energy above Coulomb
barrier brings enough excitation energy
to evaporate several nucleons
 5-10 MeV deexcitation for neutron
evaporation
• heavy-ion reactions needed for
reaching predicted island of stability
around Z=114 to N=184
• U is excitation energy, MA and Ma
masses of target and projectile, Ta is
projectile kinetic energy, Sa is projectile
binding energy in compound nucleus
MA
U
Ta  S a
M A  Ma
3-28
3-29
Photonuclear reactions
• Reactions between nuclei and lowand medium-energy photons
dominated by giant resonance

Excitation function for
photon absorption goes
through a broad maximum a
few MeV wide
 Due to excitation of
dipole vibrations of
protons against neutrons
in the nucleus
• Resonance peak varies smoothly
with A

24 MeV at 16O

13 MeV at 209Bi
• Peak cross sections are 100-300 mb
• (g, p), (g, n), (g,) reactions
http://www.engin.umich.edu/research
/cuos/ResearchGroups/HFS/Research
/photonuclear_reactions.html
3-30
Natural Element Production
•
•
•
•
•
•
Nuclear Astrophysics

fundamental information on the properties of nuclei and their reactions
to the

perceived properties of astrological objects

processes that occur in space
Nuclear reactions responsible for production of elements

Occurs in stars
At temperatures and densities

light elements are ionized and have high enough thermal velocities to
induce a nuclear reaction

heavier elements were created by a variety of nuclear processes in
massive stellar systems
systems must explode to disperse the heavy elements

distribution of isotopes here on earth
underlying information on the elemental abundances
nuclear processes to produce the primordial elements
3-31
•Chart of the
nuclide
trends
•Actinides
some
distance
from stable
elements
3-32
Timeline
• Big bang 15E9 years
ago
• Temperature 1E9 K
• Upon cooling influence
of forces felt

2 hours
 H (89 %)
and He (11
%)

Free neutrons
decay
3-33
Origin of Elements
•
•
•
•
Gravitational coalescence of H and He into clouds
Increase in temperature to fusion
Proton reaction
1H + n → 2H + g

2H + 1H → 3He

2H + n → 3H

3H + 1H → 4He + g

3He + n → 4He + g

3H + 2H → 4He + n

2H + 2H → 4He + g

4He + 3H → 7Li + g

3He+4He → 7Be + g

 7Be short lived
 Initial nucleosynthesis lasted 30 minutes
* Consider neutron reaction and free neutron half life
Further nucleosynthesis in stars

No EC process in stars
3-34
Stellar
Nucleosynthesis
•
He burning
4He + 4He ↔ 8Be +

γ - 91.78 keV
 Too short
lived

3 4He → 12C + γ +
7.367 MeV
12C + 4He →16O

16O + 4He →20Ne

•
Formation of 12C based
on Hoyle state

Excited nuclear
state
 Somewhat
different
from ground
state 12C

Around 7.6 MeV
above ground
state

0+
Fusion up to Fe

From binding
energy curve

Maximum at Fe
•
3-35
Stellar Nucleosynthesis
•
CNO cycle
12C + 1H →13N +

g
13N →13C + e++ νe

13C + 1H →14N +

γ
14N + 1H →15O +

γ
15O →15N + e+ +

νe
15N + 1H →12C +

4He

Net result is
conversion of 4
protons to alpha
particle
 4 1H → 4He
+2 e++ 2 νe
+3 γ
3-36
Formation of elements A>60
Neutron Capture; S-process

A>60
68Zn(n, γ) 69Zn, 69Zn → 69Ga +   n


mean times of neutron capture reactions longer than beta decay
half-life
 Isotope can beta decay before another capture

Up to Bi
3-37
Nucleosynthesis: R process
• Neutron capture time scale very much less than - decay lifetimes
• Neutron density 1028/m3

Extremely high flux

capture times of the order of fractions of a second

Unstable neutron rich nuclei
• rapidly decay to form stable neutron rich nuclei
• all A<209 and peaks at N=50,82, 126 (magic numbers)
3-38
•
•
•
•
•
•
P process
Formation of proton rich nuclei
Proton capture process
70<A<200
Photonuclear process, at higher Z (around 40)

(g, p), (g,), (g, n)
190Pt and 168Yb from p process

Also associated with proton capture process (p,g)
Variation on description in the literature
3-39
• Proton-rich nuclei
with Z = 7-26
 Forms a small
number of
nuclei with A<
100
• (p,g) and + decays
that populate the prich nuclei
 Also associated
with rapid
proton capture
process
• Initiates as a side
chain of the CNO
cycle
 21Na and 19Ne
rp process
(rapid proton capture)
3-40
Review Notes
• Understand Reaction Notation
• Understand Energetics of Nuclear Reactions
 Q values and barriers
• Understand the Different Reaction Types and
Mechanisms
 Particles
 Energy
• Relate cross sections to energy
• Describe Photonuclear Reactions
• Routes and reactions in nucleosynthesis
• Influence of reaction rate and particles on
nucleosynthesis
3-41
Questions
• Describe the different types of nuclear reactions for heavy ions.
• Provide notations for the following

Reaction of 16O with 208Pb to make stable Au

Formation of Pu from Th and a projectile
• Find the threshold energy for the reaction of 59Co and an alpha
that produces a neutron and a product nuclei
• What are the differences between low and high energy reactions?
• How does a charged particle reaction change with energy? A
neutron reaction?
• How are actinides made in nucleosynthesis?
• What is the s-process?
• What elements were produced in the big bang?
• Which isotopes are produced by photonuclear reactions?
• What is interesting about the production of 12C
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Question
• Respond to PDF Quiz
• Comment in blog
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